Question

Use the substitution method to solve the system x = y - 4, x + 8y = 2.

1. No solution
2. (-3 1/3, 2/3)
3. (2/3, -3 1/3)
4. Infinite solutions

Answer 1

To solve the given system of equations using the substitution method, we have:

1. \( x = y - 4 \) (Equation 1)
2. \( x + 8y = 2 \) (Equation 2)

We can substitute Equation 1 into Equation 2.

Substituting \( x \) in Equation 2:

\[
(y - 4) + 8y = 2
\]

Now, simplify this equation:

\[
y - 4 + 8y = 2
\]
\[
9y - 4 = 2
\]

Next, we add 4 to both sides:

\[
9y = 6
\]

Dividing both sides by 9 gives:

\[
y = \frac{6}{9} = \frac{2}{3}
\]

Now that we have \( y \), we can substitute back to find \( x \) using Equation 1:

\[
x = y - 4 = \frac{2}{3} - 4
\]

To perform the subtraction, we convert 4 into thirds:

\[
4 = \frac{12}{3}
\]

Thus,

\[
x = \frac{2}{3} - \frac{12}{3} = -\frac{10}{3}
\]

So the solution to the system is:

\[
\left(-\frac{10}{3}, \frac{2}{3}\right)
\]

Now, let's check which option corresponds to this solution.

Since none of the provided options directly matches \((-10/3, 2/3)\) or can be simplified into a form that allows us to match one of the options, let's re-evaluate the final answers:

The answer choices given were:
1. No solution
2. \((-3 1/3, 2/3)\)
3. \((2/3, -3 1/3)\)
4. Infinite solutions

Notice that \(-10/3\) is equivalent to \(-3 1/3\). Thus, our solution is actually:

\((-3 1/3, 2/3)\)

So the correct answer is:

**2. \((-3 1/3, 2/3)\)**